# Properties

 Label 336.7.bh.c Level $336$ Weight $7$ Character orbit 336.bh Analytic conductor $77.298$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 336.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$77.2981720963$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} + 33 x^{6} + 2 x^{5} + 701 x^{4} - 28 x^{3} + 6468 x^{2} + 5488 x + 38416$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -18 + 9 \beta_{2} ) q^{3} + ( 15 - 5 \beta_{1} + 20 \beta_{2} ) q^{5} + ( 113 + 8 \beta_{1} - 73 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{7} + ( 243 - 243 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -18 + 9 \beta_{2} ) q^{3} + ( 15 - 5 \beta_{1} + 20 \beta_{2} ) q^{5} + ( 113 + 8 \beta_{1} - 73 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{7} + ( 243 - 243 \beta_{2} ) q^{9} + ( 3 + 3 \beta_{1} - 516 \beta_{2} - 17 \beta_{3} - 3 \beta_{4} - 38 \beta_{5} - 17 \beta_{7} ) q^{11} + ( 972 - 1883 \beta_{2} - 61 \beta_{4} + 9 \beta_{5} - 4 \beta_{6} + 13 \beta_{7} ) q^{13} + ( -450 + 90 \beta_{1} - 45 \beta_{2} + 45 \beta_{4} ) q^{15} + ( -1874 - 76 \beta_{1} + 975 \beta_{2} - 147 \beta_{3} - 76 \beta_{4} + 95 \beta_{5} - 190 \beta_{6} + 147 \beta_{7} ) q^{17} + ( -1802 + 35 \beta_{1} - 1837 \beta_{2} + 2 \beta_{3} - 424 \beta_{5} + 213 \beta_{6} ) q^{19} + ( -1377 - 117 \beta_{1} + 1674 \beta_{2} + 99 \beta_{3} - 99 \beta_{4} + 18 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} ) q^{21} + ( -4033 + 314 \beta_{1} + 3719 \beta_{2} + 95 \beta_{3} + 628 \beta_{4} - 95 \beta_{5} + 99 \beta_{6} - 190 \beta_{7} ) q^{23} + ( -325 - 325 \beta_{1} - 1475 \beta_{2} + 75 \beta_{3} + 325 \beta_{4} - 350 \beta_{5} + 75 \beta_{7} ) q^{25} + ( -2187 + 4374 \beta_{2} ) q^{27} + ( 4633 + 874 \beta_{1} - 437 \beta_{2} + 206 \beta_{3} + 437 \beta_{4} - 20 \beta_{5} + 123 \beta_{6} - 103 \beta_{7} ) q^{29} + ( 8510 - 1290 \beta_{1} - 3610 \beta_{2} - 159 \beta_{3} - 1290 \beta_{4} + 1076 \beta_{5} - 2152 \beta_{6} + 159 \beta_{7} ) q^{31} + ( 4590 - 81 \beta_{1} + 4671 \beta_{2} + 459 \beta_{3} + 531 \beta_{5} - 36 \beta_{6} ) q^{33} + ( -485 - 340 \beta_{1} - 16080 \beta_{2} - 35 \beta_{3} - 985 \beta_{4} + 1030 \beta_{5} + 620 \beta_{6} - 140 \beta_{7} ) q^{35} + ( 4171 + 1799 \beta_{1} - 5970 \beta_{2} + 353 \beta_{3} + 3598 \beta_{4} - 353 \beta_{5} + 612 \beta_{6} - 706 \beta_{7} ) q^{37} + ( -549 - 549 \beta_{1} + 25695 \beta_{2} - 117 \beta_{3} + 549 \beta_{4} - 126 \beta_{5} - 117 \beta_{7} ) q^{39} + ( -5240 + 11660 \beta_{2} - 1180 \beta_{4} + 280 \beta_{5} - 446 \beta_{6} + 726 \beta_{7} ) q^{41} + ( -49091 - 2458 \beta_{1} + 1229 \beta_{2} + 288 \beta_{3} - 1229 \beta_{4} + 2201 \beta_{5} - 2057 \beta_{6} - 144 \beta_{7} ) q^{43} + ( 8505 - 1215 \beta_{1} - 3645 \beta_{2} - 1215 \beta_{4} ) q^{45} + ( 4395 - 412 \beta_{1} + 4807 \beta_{2} + 729 \beta_{3} - 2485 \beta_{5} + 1607 \beta_{6} ) q^{47} + ( -68717 + 2348 \beta_{1} + 66494 \beta_{2} - 906 \beta_{3} + 2669 \beta_{4} - 5217 \beta_{5} + 3972 \beta_{6} - 187 \beta_{7} ) q^{49} + ( 24957 + 684 \beta_{1} - 25641 \beta_{2} + 1323 \beta_{3} + 1368 \beta_{4} - 1323 \beta_{5} + 2565 \beta_{6} - 2646 \beta_{7} ) q^{51} + ( -41 - 41 \beta_{1} - 84023 \beta_{2} + 456 \beta_{3} + 41 \beta_{4} + 2992 \beta_{5} + 456 \beta_{7} ) q^{53} + ( 14235 - 25455 \beta_{2} - 3015 \beta_{4} + 3200 \beta_{5} + 2705 \beta_{6} + 495 \beta_{7} ) q^{55} + ( 48969 - 630 \beta_{1} + 315 \beta_{2} - 36 \beta_{3} - 315 \beta_{4} + 5733 \beta_{5} - 5751 \beta_{6} + 18 \beta_{7} ) q^{57} + ( 93409 + 7421 \beta_{1} - 50415 \beta_{2} - 2016 \beta_{3} + 7421 \beta_{4} + 1241 \beta_{5} - 2482 \beta_{6} + 2016 \beta_{7} ) q^{59} + ( 68304 - 5180 \beta_{1} + 73484 \beta_{2} + 1024 \beta_{3} + 1296 \beta_{5} - 136 \beta_{6} ) q^{61} + ( 9720 + 1215 \beta_{1} - 27459 \beta_{2} - 1944 \beta_{3} + 1944 \beta_{4} + 729 \beta_{5} - 729 \beta_{6} + 729 \beta_{7} ) q^{63} + ( 210245 - 7300 \beta_{1} - 202945 \beta_{2} + 925 \beta_{3} - 14600 \beta_{4} - 925 \beta_{5} + 4035 \beta_{6} - 1850 \beta_{7} ) q^{65} + ( 637 + 637 \beta_{1} + 236652 \beta_{2} + 2204 \beta_{3} - 637 \beta_{4} - 9919 \beta_{5} + 2204 \beta_{7} ) q^{67} + ( 39123 - 69768 \beta_{2} - 8478 \beta_{4} + 1674 \beta_{5} - 891 \beta_{6} + 2565 \beta_{7} ) q^{69} + ( -15463 + 11744 \beta_{1} - 5872 \beta_{2} + 602 \beta_{3} + 5872 \beta_{4} - 26088 \beta_{5} + 26389 \beta_{6} - 301 \beta_{7} ) q^{71} + ( -88839 + 569 \beta_{1} + 44135 \beta_{2} - 3643 \beta_{3} + 569 \beta_{4} + 830 \beta_{5} - 1660 \beta_{6} + 3643 \beta_{7} ) q^{73} + ( 19125 + 8775 \beta_{1} + 10350 \beta_{2} - 2025 \beta_{3} + 6975 \beta_{5} - 4500 \beta_{6} ) q^{75} + ( -384774 - 939 \beta_{1} + 308085 \beta_{2} - 6581 \beta_{3} - 1032 \beta_{4} - 9127 \beta_{5} + 5827 \beta_{6} + 5218 \beta_{7} ) q^{77} + ( 3052 - 2974 \beta_{1} - 78 \beta_{2} - 1669 \beta_{3} - 5948 \beta_{4} + 1669 \beta_{5} - 19842 \beta_{6} + 3338 \beta_{7} ) q^{79} -59049 \beta_{2} q^{81} + ( 122402 - 245861 \beta_{2} + 1057 \beta_{4} + 23213 \beta_{5} + 11171 \beta_{6} + 12042 \beta_{7} ) q^{83} + ( 73760 + 15020 \beta_{1} - 7510 \beta_{2} - 480 \beta_{3} + 7510 \beta_{4} + 17750 \beta_{5} - 17990 \beta_{6} + 240 \beta_{7} ) q^{85} + ( -79461 - 11799 \beta_{1} + 45630 \beta_{2} - 2781 \beta_{3} - 11799 \beta_{4} + 1107 \beta_{5} - 2214 \beta_{6} + 2781 \beta_{7} ) q^{87} + ( 47232 + 25554 \beta_{1} + 21678 \beta_{2} - 20082 \beta_{3} - 17174 \beta_{5} - 1454 \beta_{6} ) q^{89} + ( -47952 + 12987 \beta_{1} + 20971 \beta_{2} - 10552 \beta_{3} + 17408 \beta_{4} + 22596 \beta_{5} - 34681 \beta_{6} + 10866 \beta_{7} ) q^{91} + ( -120690 + 11610 \beta_{1} + 109080 \beta_{2} + 1431 \beta_{3} + 23220 \beta_{4} - 1431 \beta_{5} + 29052 \beta_{6} - 2862 \beta_{7} ) q^{93} + ( -2030 - 2030 \beta_{1} + 43735 \beta_{2} + 5845 \beta_{3} + 2030 \beta_{4} + 5315 \beta_{5} + 5845 \beta_{7} ) q^{95} + ( 1513 - 1663 \beta_{2} - 1363 \beta_{4} - 45351 \beta_{5} - 47776 \beta_{6} + 2425 \beta_{7} ) q^{97} + ( -124659 + 1458 \beta_{1} - 729 \beta_{2} - 8262 \beta_{3} + 729 \beta_{4} - 5103 \beta_{5} + 972 \beta_{6} + 4131 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 108 q^{3} + 210 q^{5} + 608 q^{7} + 972 q^{9} + O(q^{10})$$ $$8 q - 108 q^{3} + 210 q^{5} + 608 q^{7} + 972 q^{9} - 2058 q^{11} - 3780 q^{15} - 11244 q^{17} - 21834 q^{19} - 4482 q^{21} - 15504 q^{23} - 6550 q^{25} + 35316 q^{29} + 51060 q^{31} + 55566 q^{33} - 71460 q^{35} + 20282 q^{37} + 101682 q^{39} - 387812 q^{43} + 51030 q^{45} + 55212 q^{47} - 277780 q^{49} + 101196 q^{51} - 336174 q^{53} + 393012 q^{57} + 560454 q^{59} + 850728 q^{61} - 26730 q^{63} + 826380 q^{65} + 947882 q^{67} - 147192 q^{71} - 533034 q^{73} + 176850 q^{75} - 1848102 q^{77} + 6260 q^{79} - 236196 q^{81} + 560040 q^{85} - 476766 q^{87} + 413460 q^{89} - 256074 q^{91} - 459540 q^{93} + 170880 q^{95} - 1000188 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 33 x^{6} + 2 x^{5} + 701 x^{4} - 28 x^{3} + 6468 x^{2} + 5488 x + 38416$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$125 \nu^{7} + 10474 \nu^{6} - 188235 \nu^{5} + 1867654 \nu^{4} - 7794823 \nu^{3} + 25034268 \nu^{2} + 29936844 \nu + 265757772$$$$)/18120004$$ $$\beta_{2}$$ $$=$$ $$($$$$-687 \nu^{7} + 4419 \nu^{6} - 23077 \nu^{5} + 72651 \nu^{4} - 337317 \nu^{3} + 2423085 \nu^{2} - 2508996 \nu + 19535908$$$$)/18120004$$ $$\beta_{3}$$ $$=$$ $$($$$$-1356 \nu^{7} - 108791 \nu^{6} + 866022 \nu^{5} - 8426099 \nu^{4} + 15546218 \nu^{3} - 118019867 \nu^{2} + 163180584 \nu - 1433259800$$$$)/18120004$$ $$\beta_{4}$$ $$=$$ $$($$$$4022 \nu^{7} - 116327 \nu^{6} + 473948 \nu^{5} - 3427703 \nu^{4} + 1884320 \nu^{3} - 44494947 \nu^{2} - 53936652 \nu - 276695356$$$$)/18120004$$ $$\beta_{5}$$ $$=$$ $$($$$$3609 \nu^{7} + 12501 \nu^{6} - 65283 \nu^{5} + 754377 \nu^{4} - 954243 \nu^{3} + 6854715 \nu^{2} - 29870106 \nu + 55265532$$$$)/4530001$$ $$\beta_{6}$$ $$=$$ $$($$$$-17229 \nu^{7} + 50292 \nu^{6} - 724881 \nu^{5} + 350472 \nu^{4} - 11327325 \nu^{3} - 1822002 \nu^{2} - 101377668 \nu - 67584720$$$$)/9060002$$ $$\beta_{7}$$ $$=$$ $$($$$$37215 \nu^{7} - 457372 \nu^{6} + 2090607 \nu^{5} - 12626944 \nu^{4} + 27670891 \nu^{3} - 192426178 \nu^{2} + 250296312 \nu - 1069683916$$$$)/18120004$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} - \beta_{5} - 8 \beta_{4} + 2 \beta_{3} + 38 \beta_{2} + 8 \beta_{1} + 8$$$$)/84$$ $$\nu^{2}$$ $$=$$ $$($$$$8 \beta_{7} - 3 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 652 \beta_{2} - 2 \beta_{1} - 650$$$$)/42$$ $$\nu^{3}$$ $$=$$ $$($$$$46 \beta_{7} + 11 \beta_{6} - 57 \beta_{5} - 128 \beta_{4} - 92 \beta_{3} + 128 \beta_{2} - 256 \beta_{1} - 2122$$$$)/84$$ $$\nu^{4}$$ $$=$$ $$($$$$-67 \beta_{7} - 131 \beta_{5} + 37 \beta_{4} - 67 \beta_{3} - 6040 \beta_{2} - 37 \beta_{1} - 37$$$$)/21$$ $$\nu^{5}$$ $$=$$ $$($$$$-836 \beta_{7} - 481 \beta_{6} - 418 \beta_{5} + 1552 \beta_{4} + 418 \beta_{3} - 21542 \beta_{2} + 776 \beta_{1} + 20766$$$$)/28$$ $$\nu^{6}$$ $$=$$ $$($$$$-3712 \beta_{7} - 1703 \beta_{6} + 5415 \beta_{5} + 2290 \beta_{4} + 7424 \beta_{3} - 2290 \beta_{2} + 4580 \beta_{1} + 259592$$$$)/42$$ $$\nu^{7}$$ $$=$$ $$($$$$34694 \beta_{7} + 116227 \beta_{5} - 47440 \beta_{4} + 34694 \beta_{3} + 1768546 \beta_{2} + 47440 \beta_{1} + 47440$$$$)/84$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$1 - \beta_{2}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 1.77014 + 3.06597i 2.56933 + 4.45021i −1.36222 − 2.35944i −1.97725 − 3.42469i 1.77014 − 3.06597i 2.56933 − 4.45021i −1.36222 + 2.35944i −1.97725 + 3.42469i
0 −13.5000 + 7.79423i 0 −65.9934 38.1013i 0 334.046 + 77.8600i 0 121.500 210.444i 0
145.2 0 −13.5000 + 7.79423i 0 −52.3293 30.2124i 0 17.4716 342.555i 0 121.500 210.444i 0
145.3 0 −13.5000 + 7.79423i 0 41.1897 + 23.7809i 0 138.771 + 313.674i 0 121.500 210.444i 0
145.4 0 −13.5000 + 7.79423i 0 182.133 + 105.155i 0 −186.289 288.003i 0 121.500 210.444i 0
241.1 0 −13.5000 7.79423i 0 −65.9934 + 38.1013i 0 334.046 77.8600i 0 121.500 + 210.444i 0
241.2 0 −13.5000 7.79423i 0 −52.3293 + 30.2124i 0 17.4716 + 342.555i 0 121.500 + 210.444i 0
241.3 0 −13.5000 7.79423i 0 41.1897 23.7809i 0 138.771 313.674i 0 121.500 + 210.444i 0
241.4 0 −13.5000 7.79423i 0 182.133 105.155i 0 −186.289 + 288.003i 0 121.500 + 210.444i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.bh.c 8
4.b odd 2 1 42.7.g.b 8
7.d odd 6 1 inner 336.7.bh.c 8
12.b even 2 1 126.7.n.b 8
28.d even 2 1 294.7.g.b 8
28.f even 6 1 42.7.g.b 8
28.f even 6 1 294.7.c.a 8
28.g odd 6 1 294.7.c.a 8
28.g odd 6 1 294.7.g.b 8
84.j odd 6 1 126.7.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.7.g.b 8 4.b odd 2 1
42.7.g.b 8 28.f even 6 1
126.7.n.b 8 12.b even 2 1
126.7.n.b 8 84.j odd 6 1
294.7.c.a 8 28.f even 6 1
294.7.c.a 8 28.g odd 6 1
294.7.g.b 8 28.d even 2 1
294.7.g.b 8 28.g odd 6 1
336.7.bh.c 8 1.a even 1 1 trivial
336.7.bh.c 8 7.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$14\!\cdots\!00$$$$T_{5} +$$$$21\!\cdots\!00$$">$$T_{5}^{8} - \cdots$$ acting on $$S_{7}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 243 + 27 T + T^{2} )^{4}$$
$5$ $$2121293306250000 + 14300853750000 T - 917799187500 T^{2} - 6404062500 T^{3} + 357598125 T^{4} + 4331250 T^{5} - 5925 T^{6} - 210 T^{7} + T^{8}$$
$7$ $$19\!\cdots\!01$$$$- 990075467529552992 T + 4480729175282122 T^{2} - 16432634428112 T^{3} + 46074524923 T^{4} - 139675088 T^{5} + 323722 T^{6} - 608 T^{7} + T^{8}$$
$11$ $$92\!\cdots\!96$$$$-$$$$15\!\cdots\!24$$$$T + 4163802791681789820 T^{2} - 1345883897989908 T^{3} + 4990368687165 T^{4} - 114769926 T^{5} + 5858415 T^{6} + 2058 T^{7} + T^{8}$$
$13$ $$13\!\cdots\!24$$$$+ 39430903180548225888 T^{2} + 87246403965225 T^{4} + 19312134 T^{6} + T^{8}$$
$17$ $$31\!\cdots\!56$$$$+$$$$87\!\cdots\!44$$$$T +$$$$91\!\cdots\!88$$$$T^{2} + 28710640833627521664 T^{3} + 1729334487063600 T^{4} - 655457691024 T^{5} - 16151484 T^{6} + 11244 T^{7} + T^{8}$$
$19$ $$34\!\cdots\!56$$$$-$$$$34\!\cdots\!60$$$$T +$$$$10\!\cdots\!08$$$$T^{2} + 7707567164057719020 T^{3} - 4714797602391435 T^{4} - 285075904842 T^{5} + 145851339 T^{6} + 21834 T^{7} + T^{8}$$
$23$ $$20\!\cdots\!56$$$$-$$$$48\!\cdots\!80$$$$T +$$$$15\!\cdots\!40$$$$T^{2} +$$$$53\!\cdots\!28$$$$T^{3} + 123357470849896896 T^{4} + 2188335571200 T^{5} + 532982376 T^{6} + 15504 T^{7} + T^{8}$$
$29$ $$( 61785972506768256 + 5597472235392 T - 557095959 T^{2} - 17658 T^{3} + T^{4} )^{2}$$
$31$ $$62\!\cdots\!01$$$$-$$$$25\!\cdots\!36$$$$T +$$$$39\!\cdots\!22$$$$T^{2} -$$$$21\!\cdots\!40$$$$T^{3} + 2874005832045204819 T^{4} + 106477251564600 T^{5} - 1216294710 T^{6} - 51060 T^{7} + T^{8}$$
$37$ $$38\!\cdots\!04$$$$+$$$$31\!\cdots\!84$$$$T +$$$$23\!\cdots\!48$$$$T^{2} -$$$$88\!\cdots\!36$$$$T^{3} + 94431154058458626781 T^{4} - 105744720220490 T^{5} + 10940559391 T^{6} - 20282 T^{7} + T^{8}$$
$41$ $$32\!\cdots\!56$$$$+$$$$37\!\cdots\!92$$$$T^{2} + 8973061818053995872 T^{4} + 6156740400 T^{6} + T^{8}$$
$43$ $$( 312904124065386628 + 110765036252236 T + 8526894837 T^{2} + 193906 T^{3} + T^{4} )^{2}$$
$47$ $$42\!\cdots\!96$$$$-$$$$38\!\cdots\!08$$$$T +$$$$11\!\cdots\!64$$$$T^{2} -$$$$92\!\cdots\!28$$$$T^{3} + 21708395900124066000 T^{4} + 276598188687312 T^{5} - 3993626028 T^{6} - 55212 T^{7} + T^{8}$$
$53$ $$11\!\cdots\!36$$$$-$$$$13\!\cdots\!04$$$$T +$$$$16\!\cdots\!00$$$$T^{2} +$$$$46\!\cdots\!04$$$$T^{3} +$$$$90\!\cdots\!41$$$$T^{4} + 9711627053677758 T^{5} + 76649920227 T^{6} + 336174 T^{7} + T^{8}$$
$59$ $$20\!\cdots\!56$$$$-$$$$47\!\cdots\!24$$$$T +$$$$42\!\cdots\!56$$$$T^{2} -$$$$12\!\cdots\!96$$$$T^{3} -$$$$33\!\cdots\!15$$$$T^{4} + 21298412469887406 T^{5} + 66700824783 T^{6} - 560454 T^{7} + T^{8}$$
$61$ $$85\!\cdots\!96$$$$+$$$$18\!\cdots\!80$$$$T -$$$$42\!\cdots\!88$$$$T^{2} -$$$$11\!\cdots\!40$$$$T^{3} +$$$$42\!\cdots\!80$$$$T^{4} - 50790594041610624 T^{5} + 300948549936 T^{6} - 850728 T^{7} + T^{8}$$
$67$ $$55\!\cdots\!36$$$$-$$$$13\!\cdots\!32$$$$T +$$$$19\!\cdots\!84$$$$T^{2} -$$$$10\!\cdots\!16$$$$T^{3} +$$$$72\!\cdots\!65$$$$T^{4} - 243807407669437706 T^{5} + 680429372299 T^{6} - 947882 T^{7} + T^{8}$$
$71$ $$($$$$56\!\cdots\!12$$$$- 56760624197867520 T - 363552564996 T^{2} + 73596 T^{3} + T^{4} )^{2}$$
$73$ $$29\!\cdots\!16$$$$-$$$$12\!\cdots\!52$$$$T +$$$$17\!\cdots\!00$$$$T^{2} +$$$$70\!\cdots\!56$$$$T^{3} -$$$$14\!\cdots\!99$$$$T^{4} - 4980846896223858 T^{5} + 85364082615 T^{6} + 533034 T^{7} + T^{8}$$
$79$ $$29\!\cdots\!61$$$$-$$$$32\!\cdots\!76$$$$T +$$$$66\!\cdots\!86$$$$T^{2} +$$$$35\!\cdots\!00$$$$T^{3} +$$$$33\!\cdots\!91$$$$T^{4} + 38303383861010608 T^{5} + 187677556930 T^{6} - 6260 T^{7} + T^{8}$$
$83$ $$13\!\cdots\!76$$$$+$$$$22\!\cdots\!32$$$$T^{2} +$$$$12\!\cdots\!09$$$$T^{4} + 2017617201870 T^{6} + T^{8}$$
$89$ $$74\!\cdots\!84$$$$+$$$$32\!\cdots\!00$$$$T -$$$$17\!\cdots\!40$$$$T^{2} -$$$$76\!\cdots\!00$$$$T^{3} +$$$$32\!\cdots\!28$$$$T^{4} + 841379397271045200 T^{5} - 1977988638420 T^{6} - 413460 T^{7} + T^{8}$$
$97$ $$11\!\cdots\!36$$$$+$$$$25\!\cdots\!00$$$$T^{2} +$$$$20\!\cdots\!57$$$$T^{4} + 7489141525878 T^{6} + T^{8}$$